Plucker matrix

A Plücker matrix is a special skew-symmetric real matrix, which characterizes a projective straight line in the four-dimensional space. Since it is based on homogeneous coordinates, it is determined up to a nonzero real factor. It is named after the German mathematician Julius Plücker. The Plücker matrix is part of the higher-dimensional Plücker coordinates.

Definition

The Plucker matrix of a straight line is defined by two points on the straight line, represented as column vectors.

Dual Pliicker matrix of a straight line defined by the intersection of two planes, and which coincides with the given line. The levels or contain all points whose scalar product with the column vectors or equal to zero.

The following relationship applies the matrix elements:

Properties

Each Plücker matrix has only rank 2 and four degrees of freedom ( as every line in ). It is independent of the choice of the points A and B, and is also a generalization of the linear equation.

Example

The X-axis may be represented by